L(s) = 1 | + 0.230·2-s − 1.04·3-s − 1.94·4-s − 0.240·6-s − 1.53·7-s − 0.909·8-s − 1.90·9-s + 1.08·11-s + 2.03·12-s − 4.04·13-s − 0.352·14-s + 3.68·16-s − 5.09·17-s − 0.439·18-s + 2.30·19-s + 1.60·21-s + 0.250·22-s + 5.69·23-s + 0.950·24-s − 0.932·26-s + 5.13·27-s + 2.98·28-s + 1.38·29-s + 8.98·31-s + 2.66·32-s − 1.13·33-s − 1.17·34-s + ⋯ |
L(s) = 1 | + 0.162·2-s − 0.603·3-s − 0.973·4-s − 0.0983·6-s − 0.578·7-s − 0.321·8-s − 0.635·9-s + 0.327·11-s + 0.587·12-s − 1.12·13-s − 0.0942·14-s + 0.921·16-s − 1.23·17-s − 0.103·18-s + 0.529·19-s + 0.349·21-s + 0.0533·22-s + 1.18·23-s + 0.194·24-s − 0.182·26-s + 0.987·27-s + 0.563·28-s + 0.256·29-s + 1.61·31-s + 0.471·32-s − 0.197·33-s − 0.201·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 241 | \( 1 - T \) |
good | 2 | \( 1 - 0.230T + 2T^{2} \) |
| 3 | \( 1 + 1.04T + 3T^{2} \) |
| 7 | \( 1 + 1.53T + 7T^{2} \) |
| 11 | \( 1 - 1.08T + 11T^{2} \) |
| 13 | \( 1 + 4.04T + 13T^{2} \) |
| 17 | \( 1 + 5.09T + 17T^{2} \) |
| 19 | \( 1 - 2.30T + 19T^{2} \) |
| 23 | \( 1 - 5.69T + 23T^{2} \) |
| 29 | \( 1 - 1.38T + 29T^{2} \) |
| 31 | \( 1 - 8.98T + 31T^{2} \) |
| 37 | \( 1 + 1.25T + 37T^{2} \) |
| 41 | \( 1 - 2.78T + 41T^{2} \) |
| 43 | \( 1 - 6.56T + 43T^{2} \) |
| 47 | \( 1 - 3.61T + 47T^{2} \) |
| 53 | \( 1 + 1.93T + 53T^{2} \) |
| 59 | \( 1 - 3.87T + 59T^{2} \) |
| 61 | \( 1 + 3.33T + 61T^{2} \) |
| 67 | \( 1 + 9.93T + 67T^{2} \) |
| 71 | \( 1 - 7.32T + 71T^{2} \) |
| 73 | \( 1 + 2.83T + 73T^{2} \) |
| 79 | \( 1 + 5.12T + 79T^{2} \) |
| 83 | \( 1 + 6.31T + 83T^{2} \) |
| 89 | \( 1 + 7.31T + 89T^{2} \) |
| 97 | \( 1 - 4.89T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.70196169103033539304731465277, −6.86344055345221719231652975510, −6.23862843136342620938822375942, −5.50474174563638698149347974565, −4.78057164801819600333101643977, −4.33672713169489714965332041567, −3.17982003679832508437143880570, −2.57835222937042156989069474574, −0.938682116649152381040439418642, 0,
0.938682116649152381040439418642, 2.57835222937042156989069474574, 3.17982003679832508437143880570, 4.33672713169489714965332041567, 4.78057164801819600333101643977, 5.50474174563638698149347974565, 6.23862843136342620938822375942, 6.86344055345221719231652975510, 7.70196169103033539304731465277